The Basic PRV is a globe-pattern valve, spring loaded to open. THe downstream pressure is connected to the diaphragm actuator, opposing the spring. Adjusting the spring compression controls the set point. When the downstream pressure is greater than the setpoint, the valve closes. When the downstream pressure is less than the setpoint, the valve opens.
Since the spring has a K-factor and the valve disc has measurable travel, Increasing flow requires the valve to stroke further to open the disc proportional to the flowrate. For this to happen the downstream pressure must deviate farther from set pressure for greater flow. This phenom is called "Droop". The PID equivalent is proportional band.
Usually the flow in a regulator tends to close the disc, and most regulators have unbalanced discs. The ratio of the disc area to the diaphragm area determines the upstream pressure effect, or inverse sympathetic ratio. If the diaphragm area is 10 times the disc area, for an increase in supply pressure of 10 psi, the setpoint will decrease by 1 psi.
PRV enhancements include dome loading (gas pressure replaces the spring; K=0), Balanced plug (Decreased inverse sympathetic ratio), and Pilot actuation (small regulator senses downstream pressure and controls a larger regulator).
PRVs give straight proportional control. There is no Reset or Rate effect. (P has a value. I and D are zero) PRVs handle mundane pressure control duties. If you need greater precision you must use a control valve with a control loop.